Biomedical Engineering Reference
In-Depth Information
In the remainder of this chapter, we will begin by giving a more detailed de-
scription of the basic level set method. Next, some of the recent modifications to
the method will be explored, particularly those relevant to the medical imaging
community. The chapter will conclude with a brief review of the myriad applica-
tions of the level set and fast marching methods that have been published over
the last few years.
4.2 Basic Level Set Method
In this section, the necessary pieces for implementing the general level set
method are presented. These include the implicit representation of the inter-
face, the equation which describes interface motion, and the gradient control
process. There are now two methods for gradient control: reinitialization and
velocity extensions. Both of these methods will require some background in-
formation on the fast marching method for implementation. The fast marching
method is an interesting method in its own right, and a description of this method
will also be presented.
4.2.1 The Level Set Representation
At the heart of the level set method is the implicit representation of the interface.
If the interface is given by , can then be represented by a function φ , called
the level set function, defined by the signed distance function
φ ( x ) d ( x ) .
(4.1)
Here d ( x ) is the distance from the point x to the interface , and the sign is
determined so that it is negative on the inside and positive on the outside. At
any time, the interface can be recovered by locating the set
={ x : φ ( x ) = 0 }≡ φ 1 (0) .
(4.2)
For example, a circle interface and the corresponding level set function repre-
sentation are shown in Fig. 4.1.
For most applications, this representation works well, but there are inter-
faces which cannot use it. For example, interfaces with triple junctions or any
interface which does not have a clearly defined inside and outside cannot easily
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